How did they determine this age? How could we possibly predict the age of any exoplanet?

I can understand estimates of size and orbit shape but age seems like a shot in the dark, errr, space.

To quote Jon Jenkins from NASA's Ames Research Center:

"It’s awe-inspiring to consider that this planet has spent 6 billion
years in the habitable zone of its star; longer than Earth. That’s
substantial opportunity for life to arise, should all the necessary
ingredients and conditions for life exist on this planet.”

2 Answers
2

The age of the exoplanet is not independently derived, but instead, taken from the age of the host star.

So how do they calculate the age of the host star? Wikipedia has a number of methods:

As stars grow older, their luminosity increases at an appreciable rate. Given the mass of the star, one can use this rate of increase in luminosity in order to determine the age of the star.

Various properties of stars can also be used to determine their age. For example, the Eta Carinae system is emitting large quantities of gas and dust. These enormous outbursts can be used to infer that the star system is nearing the end of its life, and will explode as a supernova within a relatively short period of astronomical time.

Gyro-chronology is a method used to determine the age of field stars by measuring their rotation rate, and then comparing this rate with the rotation rate of the Sun, which serves as a precalibrated clock for this measurement.

Membership in a star cluster or star system permits an assignment of rough ages to a large number of stars present within. When one can determine the age of stars through other methods, such as the ones listed above, one can identify the age of all of the bodies in a system.

Along with other factors, the presence of a protoplanetary disk sets a maximum limit on the age of stars. Stars with protoplanetary disks are typically young, having moved onto the main sequence only a relatively short time ago. This likely isn't applicable for your question considering you're asking about planets that would be formed from the disk.

How does one determine the age of anything?
Generally humans use experience of known items to guess the age of something which they deem similar to what they have encountered before. Or else they consult an eye-witness to the event of birth to verify age as in using a birth certificate.
In the case of a planet out in space, scientists would use the age of what is known, i.e. our own solar system as some kind of calibration clock.
The assumptions would be that the object under scrutiny has undergone a similar kind of history as our own planet(s) or star.
However, therein lies the big rub. The accepted theory of star birth and planetary accretion has zero credibility when it comes to actual real, documented observed evidence to support it. Current theory cannot explain how the first stars got their existence. Nor can planets grow by rocks smashing into each other - physics 101 shows clearly that they simply bounce off each other. If someone proposes a cloud of hot uranium sludge as the substrate they have to explain where that came from in the first place. And so the shortcomings go on and on.
So even though scientists have used various means to guesstimate the age of our system[e.g. radioactivity of elements in meteorites etc.], they lack real life confirmation that that age is in fact correct, i.e. they've never observed how a star or planet gets born.
In human terms, there were no eye-witnesses to confirm on a birth certificate that the birthdate is correct. This is the only true way to determine the age of anything - some record of birth plus the history that led up to the current state.
To answer the question - they cannot in any way or fashion supply a true age for extra solar planets.